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| FUNDAMENTAL REGIONS FOR CERTAIN — 
ae ce FINITE GROUPS IN ae : 


‘ 


A THESIS 


fae "PRESENTED TO THE FACULTY OF THE GRADUATE SCHOOL IN- 
/> PARTIAL FULFILMENT OF THE REQUIREMENTS FOR 
_ THE DEGREE OF DOCTOR OF PHILOSOPHY 


PACERS ey ee 
HENRY FERRIS PRICE __ 


Ae Reprinted from 

" . AmEricaN JOURNAL OF MATHEMATICS | 

. o, i Nol SSN d oe 
January, 1918 








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University of Pennspfoania 


FUNDAMENTAL REGIONS FOR CERTAIN 
FINITE GROUPS IN S4 


A THESIS 


PRESENTED TO THE FACULTY OF THE GRADUATE SCHOOL IN 
PARTIAL FULFILMENT OF THE REQUIREMENTS FOR 
THE DEGREE OF DOCTOR OF PHILOSOPHY 


BY 
HENRY FERRIS PRICE 


Reprinted from 

AMERICAN JOURNAL OF MATHEMATICS 
Vob aA. No. oJ 
January, 1918 


Fundamental Regions for Certain Finite Groups in Sa. 


By Henry F. Price. 


One of the most interesting results of the study of transformations is 
what Klein has termed the “fundamental region.” 

A fundamental region for a group of transformations is a system of points 
which contains one and only one point of every conjugate set. 

Fundamental regions in the complex plane have been studied for some 
time and are well known. Klein®* and his followers have developed the subject 
to a considerable extent. There is a close relationship between this subject 
and the elliptic modular functions and the reduction of quadratic forms. 

The fundamental regions for groups in more than one complex variable 
have not been studied much. However, J. W. Young,f in a recent paper, 
obtained such regions for cyclic groups in two complex variables. 

In this paper will be considered fundamental regions for certain finite 
groups in two complex variables. The octahedral and icosahedral groups will 
be dealt with. 

The fundamental regions for these groups in the real plane can be readily 
determined and found to be triangles bounded by the axes of reflections. In 
the case of the complex plane the problem is solved by using Hermitian forms 
which meet the real plane in the sides of these triangles. 

The problem will be solved completely in the case of the octahedral group: 
In the case of the icosahedral group it will be solved except for the points 
which reduce one or more of the Hermitian forms to zero. 

The writer wishes to express his appreciation of the assistance and 
encouragement given him by Dr. H. H. Mitchell, of the University of Pennsyl- 
vania, in the preparation of this paper. 

The ternary collineation group G,, can be generated by the following three 
operations: E,[—&,, &, &], Eel&,&, &] and E3[&,, &, &]. 

It permutes the points of the real plane. As it contains nine operations 
of order 2, there are nine reflections. As the group is simply isomorphic with 
the symmetric group on four letters, it is evident that these reflections are in 








* Felix Klein, “ Elliptischen Modulfunktionen, Vol. I, pp. 183-207. 
7+ J. W. Young, “ Fundamental Regions for Cyclical Groups of Linear Fractional Transformations 
on Two Complex Variables,” Bull. Amer. Math. Soc., Vol. XVII, p. 340. 


19 Oe 18 Maize. ty 


Prick: Fundamental Regions for Certain Finite Groups in S,. 109 


two conjugate sets. The axes of the reflections, &:=+&,, &=+8, Ro=+th, 
£,=0, &,=0 and &,=0, divide the plane into twenty-four triangles. 

Any point in one of these triangles can be transformed, by a suitably 
chosen operation of the group, into a point in any other triangle. Any triangle 
is then a fundamental region in the plane for the group G,,. 

The group permutes the complex points of the plane also. If we consider 
the totality of points in the plane, complex as well as real, as real points in 
four-space, we may ask the question whether fundamental regions exist in S, 
for the group under consideration. If one does exist it must contain one 
triangle, and only one, of the real plane. The fixed points of the transforma- 
tions would lie on the boundaries of such a fundamental region. 


Es e nat) g 
Consider the Hermitian forms £,,—£.6, and £616: in which > =2+tu 
4&3 


and bs =y-+iv. Under the G,, we have two conjugate sets of three forms 


Es 


a (1) BE—£8,, (4) Be +E, 
(2) ake —Esks and (5) EE +E Es, 
(3) E.f—E,E,, (6) EE, +Ese,. 


It is evident that there is at least one relation between the forms, i. e., 
(1) + (2) + (3) =0. 

If we consider the portion of S,in which the signs of (1), (38), (4), (5) 
and (6) are all + and make use of the relation (1)+(2)+(3)=0, we see 
that the sign of (2) is determined as —. 

This region will be written [+—+,-+-+-+] where the signs of the six 
forms are written in order. 

We shall next consider into how many such regions S, is divided by the 
six Hermitian forms. 

The forms (1), (2) and (3) are conjugate under G,,, and because of the 
relation (1)+(2)+(3)=0 admit at most six arrangements of sign. The 
forms (4), (5) and (6) are also conjugate under G,,, and admit at most eight 
arrangements of sign. There are therefore forty-eight possible choices of sign 
for (1), (2)....(6). But the eight arrangements divide into two complete 
conjugate sets of four under G,, according as the number of + signs is odd or 
even. For one of such four choices for the forms (4), (5), (6), the six 
arrangements of sign for (1), (2), (8) all are conjugate, e. g., the +-+-+ 
choice is unaltered by the group G, of permutations of the variables, and the 
G, permutes all arrangements for (1), (2), (8). 


110 Price: Fundamental Regions for Certain Finite Groups in S,. 


Hence the forty-eight possible choices of sign or possible regions in S, 
divide into two conjugate sets of twenty-four each, and two of these regions, 
one from each set, constitute a fundamental region in S, for Gy. 

The region [+—+, +++] belongs to one of the sets. By changing the 
sign of (5) to — we obtain a region [+—+, +—+] of the other set. 

Taking these two regions together we obtain T=[+—-+,-+2+] which 
is a fundamental region in S, for the group Gy, except for the points which 
reduce one or more of the Hermitian forms to zero. 

If we consider the points which reduce one or more of the forms to zero 
we are dealing with what we may call the “boundaries” of the fundamental 
regions. 

By placing the six forms in LT equal to zero singly, in pairs, in groups of 
three, etc., in all possible ways, and discarding those which are conjugates of 
others, it is found that there are twenty-three sets of points which are sections 
of I’s boundaries and which should be taken in the fundamental region for the 
group. I can be defined completely, therefore, by the sets of points: 


[+—+, +++], 
[+—+,+0+], [0—+,+++], [+—0,++4],. [+—-+4+, 0+4], 
[+—+,++0], [0—-+,+0+], [+—0,+0+], [+—+4+, 00 0], 
[+—+,0+0], [0—+,0++], [+—0,++0], [000,++4], 
[+—+,00+], [0—+,00+], [+—0,00+], [000, +0 +}, 
[+—+,+00], [0—+,+00], [+—0,0+0], [000, 00+}, 


[o—-, DESO, 


[ane 0S 0s0 nN eT 


The ternary collineation group G,, furnishes a more complex fundamental 


region in S, than G,, does. 


by the three operations: 


E,[&, ee E,] ; 


El, oecas arg be 


and 
i= E,—ake-+(a+1)é, 
Bs4 beak + (atl) estés, 
"A Es= (a+1)8:+8.—abs, 
eal a 
where «= sen Es oe a 


fond 


This group permutes the points of the real plane. 
operations of order 2, there are fifteen axes of reflections. 


It is well known that this group can be generated 


As it contains fifteen 
These lines divide 











* H. H. Mitchell, “ Determination of the Ordinary and Modular Ternary Linear Groupe,” Trans.- 
Amer. Math. Soc., Vol. XII, No. 2, p. 223. 


Price: Fundamental Regions for Certain Finite Groups in S,. 111 


the plane into sixty triangles. The intersections of these fixed lines are real 
fixed points of three classes; first, the points left invariant under the fifteen 
subgroups of order 4; second, the points invariant under the ten subgroups of 
order 6; and third, the points invariant under the six subgroups of order 10. 

Each of the sixty triangles into which the plane is divided by the fifteen 
fixed lines has for its vertices one of each of the three classes of fixed points. 

Kach of the sixty triangles is a fundamental region in the plane. The 
group also permutes the complex points of the plane. Just as in the case of 
the G,, we can consider the totality of real and complex points in the plane as 
real points in S, and seek a fundamental region for the group Gg in the higher 
space. 

Consider the Hermitian form 2£,£,+22,¢,. Under G,, there is a single set 
of fifteen forms conjugate to 22,£,+2,¢,, which can be expressed in terms of 
six forms: 


Fi\= (a —1)& 2+ (a +2) E,— (2a+1) Ff +328, +328, 
Fy= (a +2)&8,—(2a+1)&.f.+ (a —1)bs6s—3856:—386,, 
F,=—(2a+1)&:6,+ (a —1)£,£,+ (a +2) £f,+3£.8,+326, 
Fy=—(2a+1) EE, + (a —1)E Eo + (a +2)Esbs—3EE.—3bobs, 
Ps= (a +2) £,2,— (2a +1) Ff, + (a —1)&6,+38,6,+386,, 
Po= (a —1)£,2,4 (a +2) fof ,— (2a+1) Ff,—3,,—38 £5. 


The fifteen forms conjugate to 28,6+22,¢, are F,—=—F,,=—F,—F,.- 
There are twenty relations between these forms F',,+F;,+/,,=0. 

It is found that the Hermitian forms F,, F'5,, F'y., 3, and F'5, meet the 
rea] plane in the sides of the triangle whose sides are x+ay—a’=0, 7=0 
and y=0. 

For any point which lies in this triangle, the signs of F,, and F;, are both 
—, while those of F,,, F',, and F's. are all +. 

Taking F,, and F';, as negative and the other three forms as positive, and 
making use of the twenty relations between the forms, it is seen that the signs 
of the remaining ten forms are determined. Therefore, none of these ten 
forms can cross the region in S, which is determined by the five forms under 
consideration. Since for this region F;>F,>F;>F,>F,>F., it can be 
written [345216]. 

Under Gg the region [345216] has sixty conjugate regions. These meet 
the real plane in distinct triangles, the fundamental regions, for the group, in 
the plane. 


112. Price: Fundamental Regions for Certain Finite Groups in S,. 


The question arises into how many such regions is S, divided by the 
forms F’;,? ‘ 

For any point in S, not on an F’;, the values of the Ff’, are all distinct, and, 
since these six values admit at most seven hundred and twenty permutations, 
the forms F’;,, have at most seven hundred and twenty arrangements of sign. 

Under G,, the seven hundred and twenty value.systems of the F; divide 
into twelve conjugate sets of sixty each, and if one value system is taken from 
each set we obtain a fundamental region. As an example of such a funda- 
mental region we give the twelve value systems determined by the inequalities 
F,>F,>F,, Fs>Fs, F;>F.>F, and F,>F,. Notwo of these are conjugates 
under G,, and therefore they comprise a fundamental region in S, for the group, 
except for the points which reduce one or more of the F',, to zero. 


ve 








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